On Scattered Convex Geometries

Kira Adaricheva; Maurice Pouzet

doi:10.1007/s11083-016-9413-0

On Scattered Convex Geometries

Kira Adaricheva, Maurice Pouzet

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

A convex geometry is a closure space satisfying the anti-exchange axiom. For several types of algebraic convex geometries we describe when the collection of closed sets is order scattered, in terms of obstructions to the semilattice of compact elements. In particular, a semilattice Ω(η), that does not appear among minimal obstructions to order-scattered algebraic modular lattices, plays a prominent role in convex geometries case. The connection to topological scatteredness is established in convex geometries of relatively convex sets.

Original language	English
Pages (from-to)	1-28
Number of pages	28
Journal	Order
DOIs	https://doi.org/10.1007/s11083-016-9413-0
Publication status	Accepted/In press - Nov 21 2016

Keywords

Algebraic lattice
Convex geometry
Lattices of relatively convex sets
Lattices of suborders
Lattices of subsemilattices
Multi-chains
Order-scattered poset
Topologically scattered lattice

ASJC Scopus subject areas

Algebra and Number Theory
Geometry and Topology
Computational Theory and Mathematics

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AB - A convex geometry is a closure space satisfying the anti-exchange axiom. For several types of algebraic convex geometries we describe when the collection of closed sets is order scattered, in terms of obstructions to the semilattice of compact elements. In particular, a semilattice Ω(η), that does not appear among minimal obstructions to order-scattered algebraic modular lattices, plays a prominent role in convex geometries case. The connection to topological scatteredness is established in convex geometries of relatively convex sets.

KW - Algebraic lattice

KW - Convex geometry

KW - Lattices of relatively convex sets

KW - Lattices of suborders

KW - Lattices of subsemilattices

KW - Multi-chains

KW - Order-scattered poset

KW - Topologically scattered lattice

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On Scattered Convex Geometries

Abstract

Keywords

ASJC Scopus subject areas

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